In Definition [ftip-00O5], choose constant expenditure \(v(t)=gh_0/\eta \) and maintenance \(m(t)=\delta h_0/\eta \). Since \(g=\eta A-\delta >0\), these are nonnegative and sum to \(Ah_0\). The stock equation gives \(h(t)=h_0\geq h_{\min }\) for all \(t\geq 0\), so the allocation is admissible on every finite interval and

At a fixed compute price \(p>0\), a service with constant physical capacity at least \(gh_0/(\eta p)\) can supply that positive compute rate indefinitely. Thus maintenance dependence and a finite bound for each deadline are compatible with infinite lifetime compute. No conclusion about unbounded capability follows without a learning model.

Consider a single productive capital stock \(K_0>0\), output \(AK\), depreciation \(\delta _K>0\) and allocation fractions \(s,\theta >0\) with \(s+\theta \leq 1\). Invest \(sAK\), spend \(\theta AK\) on training, and allocate the remainder to other uses. If \(A>0\) and \(\gamma =sA-\delta _K>0\), then

These identities follow by solving the linear stock equation and integrating output. With service price \(p>0\) and sufficient installed service capacity, division by \(p\) gives affordable compute. For \(T>0\), this expenditure exceeds the frozen-capital estimate \(\theta AK_0T\), since \(e^{\gamma T}-1>\gamma T\). That estimate cannot bound this policy. The model assumes investment converts into usable capital without delay; construction lags and essential complements require additional states.

Aghion, Jones and Jones study AI and economic growth with production and idea-generation mechanisms. Caballero analyzes a richer financing and capital-installation mechanism with alternative long-run outcomes. Neither citation makes the exponential path here an empirical forecast.

Let \(D(t)\) denote effective task-relevant coverage, rather than raw token count. Suppose useful new human input arrives at rate \(h\geq 0\), synthetic generation at rate \(v\geq 0\) has effective yield \(a\geq 0\), and coverage depreciates at rate \(\delta _D>0\). Under the stipulated law

Direct differentiation verifies the solution. For a required coverage \(D_{\min }>0\), if \(D_0\geq D_{\min }\) and \(h+av\geq \delta _DD_{\min }\), then \(D(t)\geq D_{\min }\) at every time. If \(h+av<\delta _DD_{\min }\) and \(D_0\) is finite, the path eventually falls below the threshold. The necessary synthetic contribution is exactly \(av\geq \max \{0,\delta _DD_{\min }-h\}\). Positive synthetic yield is necessary when human inflow leaves a deficit; an affordable schedule must also produce and check the generated material.

The scalar law omits distributional coverage and estimation error. The contrast between recursive replacement in Shumailov et al., accumulation in Gerstgrasser et al., and consistency conditions in Barzilai and Shamir is a reason to specify \(a\) and the learning process, not to assume that every generated token has fixed positive knowledge value.

Let \(H,M\geq 0\) denote usable human and machine expertise. Under perfect substitution, effective expertise is \(E=H+\chi M\) with \(\chi >0\). The productive requirement \(E\geq E_{\min }>0\) is satisfied with \(H=0\) whenever \(M\geq E_{\min }/\chi \). Under essential complementarity, take \(E=\min \{H,\chi M\}\) instead. Then \(E\geq E_{\min }\) implies \(H\geq E_{\min }\). Both claims follow directly from the definitions.

A human-stock floor can therefore represent either an explicit social constraint or an indispensable productive input. The latter interpretation requires a complementarity premise. An economic model does not prove biological exclusivity merely by naming one coordinate human expertise. The organization and substitution of knowledge also depend on communication and access, as modeled by Ide and Talamàs.

Fix maintenance \(m\geq 0\), assistance intensity \(a\geq 0\), and parameters \(\eta ,\delta _0>0\), \(\xi ,\delta _1\geq 0\). Consider

Here \(\xi a\) models learning produced by assistance and \(\delta _1aH\) models lost practice. The solution is \(H(t)=H_*+(H_0-H_*)e^{-(\delta _0+\delta _1a)t}\). If \(H_0\geq H_{\min }\), the stock stays above that floor for every \(t\geq 0\) exactly when \(H_*\geq H_{\min }\); if the stationary stock is smaller, it eventually crosses below. For a finite deadline \(T\), monotonicity instead makes viability equivalent to \(H(T)\geq H_{\min }\), which may hold even when the stationary stock is smaller. Moreover,

Differentiation shows that the sign depends on the stated parameters. Neither automatic deskilling nor automatic skill improvement follows from assistance alone. Bastani et al. study learning outcomes under different assistance designs; their particular experiment does not identify universal coefficients for this stock law. Assistance and maintenance costs must also fit the resource account.

Fix one admitted assisted campaign with final score \(Z\in [0,1]\). Suppose an autonomous campaign is admitted under the same external resource caps and evaluation convention, with simulation, development and recipient learning fully charged. If its joint law of transcript, retained artifact and fresh evaluation agrees with that of the assisted campaign, then its expected acquisition score is identical.

More generally, if these laws have total variation distance at most \(\varepsilon \), where \(\operatorname {TV}(P,Q)=\sup _A|P(A)-Q(A)|\), then \(|\mathbb E_PZ-\mathbb E_QZ|\leq \varepsilon \). Indeed, \(\mathbb E_PZ=\int _0^1P(Z>t)\,dt\), and the probability difference in each integrand is at most \(\varepsilon \). Thus the autonomous score is at least \(Q_{\rm acq}^{\rm assisted}-\varepsilon \).

Approximate agreement of output laws does not establish an almost-sure resource cap: admission of the autonomous implementation is a separate premise. Nor does computability establish affordable reconstruction of the contributor's development and observations. This is the economic version of the reconstruction boundary; a claimed separation must exclude such an affordable implementation by an actual lower bound.